Proposition 3.2.4 (Functoriality of K0 for unital C-star Algebras)
- For each unital -algebras, , .
- If and are unital -algebras, and if and are -homomorphisms, then
- .
- For every pair of -algebras and , .
Parts (1) and (2) say that is a functor from the category of unital -algebras to the category of Abelian groups. Parts (3) and (4) say that maps zero objects to zero objects, and zero morphisms to zero morphisms. WE INCLUDE THE ZERO C-ALGEBRA IN THE RANKS OF UNITAL C-ALGEBRAS!
Proof:
Use the definition of the functor
- Take a map at the level of the algebras, , a * -homomorphism.
- Induce a map on the matrix algebras of each one, .
- Recall that it sends projections to projections and so it induces another map .
- Lastly define a map from which after being put inside of the grothendieck map has the same properties as the universal property of as seen in Proposition 3.1.8 (Universal property of K0).
- This forces it to factor through a unique map we denote as
So now we check that .
As desired.